Stability in Terms of Two Measures for Nonlinear Impulsive Systems on Time Scales

Abstract

We investigate some stability problems in terms of two measures for nonlinear dynamic systems on time scales with fixed moments of impulsive effects. Sufficient conditions for (uniform) stability, (uniform) asymptotic stability, and instability in terms of two measures are derived by using the method of Lyapunov functions. Our results include the existing results as special cases when the time scale reduces to the set of real numbers. Particularly, our results provide stability criteria for impulsive discrete systems in terms of two measures, which have not been investigated extensively. Two examples are presented to illustrate the efficiency of the proposed results.

1. Introduction

It is well known that the theory of impulsive differential equations provides a general framework for mathematical modeling of many real world phenomena [1, 2]. In particular, it serves as an adequate mathematical tool for studying evolution processes that are subjected to abrupt changes in their states. At the present time, the qualitative theory of such equations has been extensively studied. Many results on the stability and boundedness of their solutions are obtained [1–4]. Due to the needs of applications, the concepts of Lyapunov stability have given rise to many new notions, for example, partial stability, conditional stability, eventual stability, practical stability, and so on. A notion which unifies and includes the above concepts of stability is the notion of stability in terms of two measures which was initialed by Movchan [5]. Since the publication of Salvadori's paper [6], this unified theory in terms of two measures became popular. For a systematic introduction to the theory of stability in terms of two measures, refer to [7].

On the other hand, a theory of time scales or calculus on measure chains was introduced by Hilger in his Ph.D. thesis [8] in 1988, with the purpose of incorporating both the existing theory of dynamic systems on continuous and discrete time scales, namely, time scale as arbitrary closed subset of real numbers, and extending the existing theory to dynamic systems on generalized hybrid (continuous/discrete) time scales. The theory of time scales recently has gained much attention and is undergoing rapid development. Recently, various work has been done on the stability problem of dynamic systems on time scales [9–14]. For more details about the theory of time scales, refer to [15–17].

Motivated by the above discussion, in this paper, we will consider the stability problems in terms of two measures for impulsive systems on time scales. Several new stability criteria and instability criteria are obtained by using the method of Lyapunov functions. As far as we know, there are very few studies on stability analysis of impulsive discrete systems in terms of two measures. Moreover, our results can be applied to impulsive systems on other time scales in addition to the set of integers and the set of real numbers.

The rest of this paper is organized as follows. In Section 2, we introduce some basic knowledge of dynamic systems on time scales. In Section 3, we formulate the problem and present several definitions of stability and instability in terms of two measures. In Section 4, several (ℎ₀,ℎ)-stability and instability criteria are established by employing the Lyapunov function approach. For illustration of our results, two examples are shown in Section 5. Finally, some conclusions are drawn in Section 6.

2. Preliminaries

In this section, we briefly introduce some basic definitions and results concerning time scales for later use.

Let ℝ be the set of real numbers, ℝ⁺ be the set of nonnegative real numbers, ℤ be the set of integers, ℤ⁺ be the set of nonnegative integers, ℕ={1,2,…}, and 𝕋 be an arbitrary nonempty closed subset of ℝ. We assume that 𝕋 is a topological space with relative topology induced from ℝ. Then, 𝕋 is called a time scale.

Definition 1. The mappings 𝜎,𝜃∶𝕋→𝕋 defined as 𝜎(𝑡){},𝜃(𝑡){}=inf𝑠∈𝕋∶𝑠>𝑡=sup𝑠∈𝕋∶𝑠<𝑡 (1) are called forward and backward jump operators, respectively.

A nonmaximal element 𝑡∈𝕋 is called right-scattered (rs) if 𝜎(𝑡)>𝑡 and right-dense (rd) if 𝜎(𝑡)=𝑡. A nonminimal element 𝑡∈𝕋 is called left-scattered (ls) if 𝜃(𝑡)<𝑡 and left-dense (ld) if 𝜃(𝑡)=𝑡. If 𝕋 has a ls maximum 𝑚, then we define 𝕋^𝑘=𝕋⧵{𝑚}, otherwise, 𝕋^𝑘=𝕋.

Definition 2. The graininess function 𝜇∶𝕋→ℝ⁺ is defined by 𝜇(𝑡)=𝜎(𝑡)−𝑡. (2)

Definition 3. For 𝑦∶𝕋→ℝ and 𝑡∈𝕋^𝑘, one defines the delta derivative 𝑦^Δ(𝑡) of 𝑦(𝑡), to be the number (when it exists) with the property that for any 𝜀>0, there is a neighborhood 𝑈 of 𝑡 (i.e., 𝑈=(𝑡−𝛿,𝑡+𝛿)⋂𝕋 for some 𝛿>0) such that 𝑦(𝜎(𝑡))−𝑦(𝑠)−𝑦^Δ(𝑡𝜎(𝑡)|𝜎(𝑡)|)(−𝑠)≤𝜀−𝑠,∀𝑠∈𝑈. (3)

A function 𝑓∶𝕋→ℝ is rd-continuous if it is continuous at rd points in 𝕋 and its left-side limits exist at ld points in 𝕋. The set of rd-continuous functions 𝑓∶𝕋→ℝ will be denoted by 𝐶_rd=𝐶_rd(𝕋,ℝ). If 𝑓 is continuous at each rd point and each ld point, 𝑓 is said to be continuous function on 𝕋. If 𝑎,𝑏∈𝕋, then one defines the interval [𝑎,𝑏] on 𝕋 by [𝑎,𝑏]∶={t∈𝕋∶𝑎≤𝑡≤𝑏}. Open intervals and half-open intervals can be defined similarly.

Definition 4. Let 𝑓∈𝐶_rd. A function 𝑔∶𝕋→ℝ is called the antiderivative of 𝑓 on 𝕋 if it is differentiable on 𝕋 and satisfies 𝑔^Δ(𝑡)=𝑓(𝑡) for all 𝑡∈𝕋. In this case, one defines ^𝑡_𝑎𝑓(𝑠)Δ𝑠=𝑔(𝑡)−𝑔(𝑎),𝑎,𝑡∈𝕋. (4)

One says that a function 𝑝∶𝕋→ℝ is regressive if 1+𝜇(𝑡)𝑝(𝑡)≠0 for all 𝑡∈𝕋. The set of all regressive and rd-continuous functions 𝑓∶𝕋→ℝ is denoted by 𝐶_rdℛ=𝐶_rdℛ(𝕋,ℝ), and the set of all positively regressive elements of 𝐶_rdℛ is denoted by 𝐶_rdℛ⁺=𝐶_rdℛ⁺(𝕋,ℝ)={𝑝∈𝐶_rdℛ∶1+𝜇(𝑡)𝑝(𝑡)>0 for all 𝑡∈𝕋}.

Definition 5. If 𝑝∈𝐶_rdℛ, then one defines the exponential function on time scale 𝕋 by 𝑒_𝑝(𝑡,𝑠)=exp^𝑡_𝑠𝜉_𝜇(𝜏)𝑝(𝜏)Δ𝜏,for𝑡,𝑠∈𝕋, (5) where the cylinder transformation 𝜉_ℎ(𝑧)=Log(1+ℎ𝑧)ℎ,ℎ≠0,𝑧,ℎ=0, (6) where Log is the principal logarithm function.

It is known that 𝑥(𝑡)=𝑒_𝑝(𝑡,𝑡₀) is the unique solution of the initial value problem 𝑥^Δ(𝑡)=𝑝(𝑡)𝑥(𝑡), 𝑥(𝑡₀)=1.

Remark 6. Let 𝛼∈𝐶_rdℛ be a constant. If 𝕋=ℤ, then 𝑒_𝛼(𝑡,𝑡₀)=(1+𝛼)^𝑡−𝑡0 for all 𝑡∈𝕋. If 𝕋=ℝ, then 𝑒_𝛼(𝑡,𝑡₀)=𝑒^{𝛼(𝑡−𝑡0)} for all 𝑡∈𝕋.

Definition 7. One says that a function 𝑚∶𝕋→ℝ is right-nondecreasing at a point 𝑡∈𝕋 provided (i)if 𝑡 is rs, then 𝑚(𝜎(𝑡))≥𝑚(𝑡);(ii)if 𝑡 is rd, then there is a neighborhood 𝑈 of 𝑡 such that 𝑚(𝑠)≥𝑚(𝑡),∀𝑠∈𝑈with𝑠>𝑡. (7)
Similarly, one says that 𝑚 is right-nonincreasing if above in (i) 𝑚(𝜎(𝑡))≤𝑚(𝑡) and in (ii) 𝑚(𝑠)≤𝑚(𝑡). If 𝑚 is right-nondecreasing (right-nonincreasing) at every 𝑡∈𝕋, one says that 𝑚 is right-nondecreasing (right-nonincreasing) on 𝕋.

Lemma 8. Let 𝑚∈𝐶(𝕋,ℝ). Then 𝑚(𝑡) is right-nondecreasing (right-nonincreasing) on 𝕋 if and only if 𝐷⁺𝑚^Δ(𝑡)≥0(𝐷⁺𝑚^Δ(𝑡)≤0) for every 𝑡∈𝕋, where 𝐷⁺𝑚^Δ(𝑡)=𝑚(𝜎(𝑡))−𝑚(𝑡)𝜇(𝑡),𝜎(𝑡)>𝑡,limsup_𝑠→𝑡+𝑚(𝑠)−𝑚(𝑡)(𝑡)𝑠−𝑡,𝜎=𝑡. (8)

Proof. The condition is obviously necessary. Let us prove that it is sufficient. We only assume 𝐷⁺𝑚^Δ(𝑡)≥0 for 𝑡∈𝕋 as the second statement can be shown similarly.
If 𝑡 is rs, then 𝐷⁺𝑚^Δ(𝑡)=𝑚(𝜎(𝑡))−𝑚(𝑡)𝜇(𝑡)≥0, (9) and hence 𝑚(𝜎(𝑡))≥𝑚(𝑡).
Let now 𝑡 to be rd, and 𝑁 be a neighborhood of 𝑡. We need to show that 𝑚(𝑠)≥𝑚(𝑡) for 𝑠>𝑡 with 𝑠∈𝑁. This follows directly from Lemma 1.1.1 in [7].
Thus the proof of the lemma is complete.

3. Problem Formulation

Consider the following nonlinear impulsive system on time scale 𝕋: 𝑥^Δ(𝑡)=𝑓(𝑡,𝑥),𝑡≠𝑡_𝑘(𝑡),𝑡∈𝕋,Δ𝑥=𝐼_𝑘(𝑥(𝑡)),𝑡=𝑡_𝑘𝑥𝑡,𝑘∈ℕ,₊₀=𝑥₀ (10) under the following assumptions.(a)𝕋 is a time scale with 𝑡₀≥0 as minimal element and no maximal element.(b){𝑡_𝑘}∈𝕋,  𝑡₀<𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯ and lim_𝑘→∞𝑡_𝑘=∞.(c)𝑥∈ℝ^𝑛 and Δ𝑥(𝑡_𝑘)=𝑥(𝑡⁺_𝑘)−𝑥(𝑡⁻_𝑘). If 𝑡_𝑘 is rd point, 𝑥(𝑡⁺_𝑘) denotes the right limit of 𝑥 at 𝑡_𝑘; if 𝑡_𝑘 is rs point, 𝑥(𝑡⁺_𝑘) denotes the state of 𝑥 at 𝑡_𝑘 with the impulse. If 𝑡_𝑘 is ld point, 𝑥(𝑡⁻_𝑘) denotes the left limit of 𝑥 at 𝑡_𝑘 with 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘) if 𝑡_𝑘 is ls point. Here, we assume that 𝑥(𝑡⁻_𝑘)=𝑥(𝑡_𝑘).(d)𝑓∶𝕋×ℝ^𝑛→ℝ^𝑛 is continuous in (𝑡_𝑘−1,𝑡_𝑘]×ℝ^𝑛 for 𝑘∈ℕ, 𝑓(𝑡,0)=0, and for each 𝑥∈ℝ^𝑛, 𝑘∈ℕ, lim_{(𝑡,𝑦)→(𝑡}⁺_𝑘,𝑥)𝑓(𝑡,𝑦)=𝑓(𝑡⁺_𝑘,𝑥);(e)𝐼_𝑘∶ℝ^𝑛→ℝ^𝑛 and 𝐼_𝑘(0)=0.

Throughout this paper, we denote by 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) the solution of system (10) satisfying initial condition 𝑥(𝑡⁺₀)=𝑥₀. Obviously, system (10) admits the trivial solution. Moreover, 𝑓 is assumed to satisfy necessary assumptions so that the following initial value problems: 𝑥^Δ=𝑓(𝑡,𝑥),𝑡∈𝑡₀,𝑡₁,𝑥𝑡₀=𝑥₀,𝑥^Δ=𝑓(𝑡,𝑥),𝑡∈𝑡_𝑘,𝑡_𝑘+1,𝑥𝑡⁺_𝑘=𝑥𝑡_𝑘+𝐼_𝑘𝑥𝑡_𝑘, (11) have unique solutions 𝑥₀(𝑡), 𝑡∈[𝑡₀,𝑡₁], and 𝑥_𝑘(𝑡), 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘∈ℕ, respectively (e.g., see [17] for existence and uniqueness results for dynamical systems on time scales.). Thus, if we define 𝑥𝑡;𝑡₀,𝑥₀=𝑥₀(𝑡),𝑡∈𝑡₀,𝑡₁𝑥₁(𝑡),𝑡∈𝑡₁,𝑡₂𝑥⋮⋮_𝑘(𝑡),𝑡∈𝑡_𝑘,𝑡_𝑘+1⋮⋮ (12) then it is easy to see that 𝑥(𝑡;𝑡₀,𝑥₀) is the unique solution of system (10).

Let us list the classes of functions and definitions for convenience.𝑃𝐶={𝛿∶𝕋→ℝ⁺, continuous on (𝑡_𝑘−1,𝑡_𝑘] and lim_𝑡→𝑡⁺_𝑘𝛿(𝑡)=𝛿(𝑡⁺_𝑘) exists for 𝑘∈ℕ}; 𝒦={𝛿∈𝐶(ℝ⁺,ℝ⁺), strictly increasing and 𝛿(0)=0}; 𝑃𝐶𝒦={𝛿∶𝕋×ℝ⁺→ℝ⁺,𝛿(⋅,𝑢)∈𝑃𝐶 for each 𝑢∈ℝ⁺ and 𝛿(𝑡,⋅)∈𝒦 for each 𝑡∈𝕋}; Γ={ℎ∶𝕋×ℝ^𝑛→ℝ⁺,ℎ(⋅,𝑥)∈𝑃𝐶 for each 𝑥∈ℝ^𝑛,ℎ(𝑡,⋅)∈𝐶(ℝ^𝑛,ℝ⁺) for each 𝑡∈𝕋 and infℎ(𝑡,𝑥)=0}; 𝜈₀={𝑉∶𝕋×ℝ^𝑛→ℝ⁺, continuous on (𝑡_𝑘−1,𝑡_𝑘]×ℝ^𝑛, 𝑘∈ℕ, and for all 𝑥∈ℝ^𝑛 and 𝑘∈ℕ, lim_{(𝑡,𝑦)→(𝑡}⁺_𝑘,𝑥)𝑉(𝑡,𝑦)=𝑣(𝑡⁺_𝑘,𝑥) exists}.

For 𝑉∈𝜈₀, (𝑡,𝑥)∈(𝑡_𝑘−1,𝑡_𝑘]×ℝ^𝑛, 𝑘∈ℕ, we define the upper right-hand Dini delta derivative of 𝑉(𝑡,𝑥) relative to (10) as follows: 𝐷⁺𝑉^Δ()=𝑉(𝜎(𝑡)(𝜎(𝑡((𝑡𝑡,𝑥,𝑥)))−𝑉𝑡,𝑥))𝜇(𝑡),𝜎(𝑡)>𝑡,limsup_𝑠→𝑡+𝑉𝑠,𝑥(𝑡)+(𝑠−𝑡)𝑓(𝑡,𝑥(𝑡))−𝑉(𝑡,𝑥(𝑡)),𝜎(𝑡)𝑠−𝑡=𝑡. (13)

Definition 9. Let ℎ₀,ℎ∈Γ. Then one says that (i) ℎ₀ is finer than ℎ if there exists a constant 𝜌>0 and a function 𝜑∈𝒦 such that ℎ₀(𝑡,𝑥)<𝜌 implies ℎ(𝑡,𝑥)≤𝜑(ℎ₀(𝑡,𝑥));(ii) ℎ₀ is weakly finer than ℎ if there exists a constant 𝜌>0 and a function 𝜑∈𝑃𝐶𝒦 such that ℎ₀(𝑡,𝑥)<𝜌 implies ℎ(𝑡,𝑥)≤𝜑(𝑡,ℎ₀(𝑡,𝑥)).

Definition 10. Let 𝑉∈𝜈₀ and ℎ₀,ℎ∈Γ. Then 𝑉(𝑡,𝑥) is said to be (i)ℎ-positive definite if there exist a 𝜌>0 and a function 𝑏∈𝒦 such that ℎ(𝑡,𝑥)<𝜌 implies 𝑏(ℎ(𝑡,𝑥))≤𝑉(𝑡,𝑥);(ii)ℎ₀-decrescent if there exist a 𝜌>0 and a function 𝑎∈𝒦 such that ℎ₀(𝑡,𝑥)<𝜌 implies 𝑉(𝑡,𝑥)≤𝑎(ℎ₀(𝑡,𝑥));(iii)ℎ₀-weakly decrescent if there exist a 𝜌>0 and a function 𝑎∈𝑃𝐶𝒦 such that ℎ₀(𝑡,𝑥)<𝜌 implies 𝑉(𝑡,𝑥)≤𝑎(𝑡,ℎ₀(𝑡,𝑥)).

Definition 11. The impulsive system (10) is said to be (S₁)  (ℎ₀,ℎ)-stable, if for each 𝜀>0, 𝑡₀∈𝕋, there exists a 𝛿=𝛿(𝑡₀,𝜀)>0 such that ℎ₀(𝑡₀,𝑥₀)<𝛿 implies ℎ(𝑡,𝑥(𝑡))<𝜀, 𝑡≥𝑡₀ for any solution 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) of (10);(S₂)  (ℎ₀,ℎ)-uniformly stable, if the 𝛿 in (S₁) is independent of 𝑡₀;(S₃)  (ℎ₀,ℎ)-attractive, if for each 𝜀>0, 𝑡₀∈𝕋, there exist two positive constants 𝛿=𝛿(𝑡₀,𝜀) and 𝑇=𝑇(𝑡₀,𝜀) such that ℎ₀(𝑡₀,𝑥₀)<𝛿 implies ℎ(𝑡,𝑥(𝑡))<𝜀, 𝑡≥𝑡₀+𝑇;(S₄)  (ℎ₀,ℎ)-uniformly attractive, if (S₃) holds with 𝛿 and 𝑇 being independent of 𝑡₀;(S₅)  (ℎ₀,ℎ)-asymptotically stable, if (S₁) and (S₃) hold simultaneously;(S₆)  (ℎ₀,ℎ)-uniformly asymptotically stable, if (S₂) and (S₄) hold together;(S₇)  (ℎ₀,ℎ)-unstable, if (S₁) fails to hold.

4. Main Results

Let us establish, in this section, sufficient conditions for (ℎ₀,ℎ)-(uniform) stability, (ℎ₀,ℎ)-(uniform) asymptotic stability, and (ℎ₀,ℎ)-instability properties of impulsive systems (10) in the following subsections, respectively. Let 𝑆ℎ,𝜌=(𝑡,𝑥)∈𝕋×ℝ^𝑛∶ℎ(𝑡,𝑥)<𝜌. (14)

4.1. (ℎ₀,ℎ)-(Uniform) Stability

Theorem 12. Assume that (i)ℎ,ℎ₀∈Γ, and ℎ₀ is weakly finer than ℎ; (ii)𝑉∈𝜈₀, 𝑉(𝑡,𝑥) is ℎ-positive definite on 𝑆(ℎ,𝜌), ℎ₀-weakly decrescent, locally Lipschtiz in 𝑥 for 𝑡∈𝕋 which is rd, and 𝐷⁺𝑉^Δ(𝑡,𝑥)≤𝑐_𝑘𝑉(),𝑡𝑡,𝑥𝑡∈_𝑘,𝑡_𝑘+1,(𝑡,𝑥)∈𝑆ℎ,𝜌, (15) where 𝑐_𝑘≥0, 𝑘∈ℤ⁺; (iii) there exists 𝜍≥𝜍_𝑘>0, such that 𝑉𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘𝑥_𝑘≤𝜍_𝑘𝑉𝑡_𝑘,𝑥_𝑘,𝑡_𝑘,𝑥_𝑘∈𝑆ℎ,𝜌, (16) where 𝑥_𝑘=𝑥(𝑡_𝑘), 𝑘∈ℕ; (iv)sup_𝑘∈ℤ⁺{∏^𝑘_𝑖=0𝜍_𝑖𝑒_𝑐𝑖(𝑡_𝑖+1,𝑡_𝑖)}=𝑀<∞ where 𝜍₀=1; (v) there exists a constant 𝜌₁, 0<𝜌₁<𝜌, such that ℎ(𝑡,𝑥)<𝜌₁ implies ℎ(𝜎(𝑡),𝑥(𝜎(𝑡)))<𝜌; (vi) there exists a constant 𝜌₀, 0<𝜌₀<𝜌₁, such that ℎ(𝑡_𝑘,𝑥_𝑘)<𝜌₀ implies ℎ(𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘(𝑥_𝑘))<𝜌₁.
 Then system (10) is (ℎ₀,ℎ)-stable.

Proof. Since 𝑉(𝑡,𝑥) is ℎ₀-weakly decrescent, there exist a constant 𝛿₀>0 and a function 𝑎∈𝑃𝐶𝒦 such that 𝑉(𝑡,𝑥)≤𝑎𝑡,ℎ₀(𝑡,𝑥),ifℎ₀(𝑡,𝑥)<𝛿₀. (17) There exists, in view of (ii), a function 𝑏∈𝒦 such that 𝑏(ℎ(𝑡,𝑥))≤𝑉(𝑡,𝑥),ifℎ(𝑡,𝑥)<𝜌. (18) By (i), there exist 𝛿₁>0 and 𝜑∈𝑃𝐶𝒦 such that ℎ(𝑡,𝑥)≤𝜑𝑡,ℎ₀(𝑡,𝑥),ifℎ₀(𝑡,𝑥)<𝛿₁. (19)
Let 𝜀∈(0,𝜌₀) and 𝑡₀∈𝕋 be given. There exists 𝛿₂=𝛿₂(𝑡₀,𝜀) such that 𝜑𝑡₀,𝛿₂<𝜌₀,max{1,𝑀}𝑎𝑡₀,𝛿₂<𝑏(𝜀). (20) Choose 𝛿=min{𝛿₀,𝛿₁,𝛿₂}. Let (𝑡₀,𝑥₀)∈𝕋×ℝ^𝑛 such that ℎ₀(𝑡₀,𝑥₀)<𝛿 and 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) be any solution of (10). Then, from (17) to (20), we get 𝑏ℎ𝑡₀,𝑥₀≤𝑉𝑡₀,𝑥₀≤𝑎𝑡₀,ℎ₀𝑡₀,𝑥₀<𝑏(𝜀), (21) which implies ℎ(𝑡₀,𝑥₀)<𝜀.
We now claim that, for every solution 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) of (10), ℎ₀(𝑡₀,𝑥₀)<𝛿 implies ℎ(𝑡,𝑥(𝑡))<𝜀,𝑡≥𝑡₀. (22) If this is not true, then there exist a solution 𝑥(𝑡) with ℎ₀(𝑡₀,𝑥₀)<𝛿 and a 𝑡^∗>𝑡₀ such that 𝑡_𝑘<𝑡^∗≤𝑡_𝑘+1, for some 𝑘, satisfying ℎ𝑡^∗,𝑥𝑡^∗≥𝜀,ℎ(𝑡,𝑥(𝑡))<𝜀for𝑡∈𝑡₀,𝑡_𝑘. (23)
Since 0<𝜀<𝜌₀, it follows form condition (vi) that ℎ𝑡⁺_𝑘,𝑥⁺_𝑘∶=ℎ𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘𝑥_𝑘<𝜌₁, (24) where 𝑥⁺_𝑘=𝑥(𝑡⁺_𝑘) and ℎ(𝑡_𝑘,𝑥_𝑘)<𝜀. Next, we will show that there exists a 𝑡⁰, 𝑡_𝑘<𝑡⁰≤𝑡^∗, such that 𝜀≤ℎ𝑡⁰,𝑥𝑡⁰<𝜌,ℎ(𝑡,𝑥(𝑡))<𝜌for𝑡∈𝑡₀,𝑡⁰. (25) To do this, we consider the following two cases:(1)there exists a 𝑡_∗, 𝑡_𝑘<𝑡_∗≤𝑡^∗, such that ℎ(𝑡_∗,𝑥(𝑡_∗))≥𝜌;(2)ℎ(𝑡,𝑥(𝑡))<𝜌 for all 𝑡∈(𝑡_𝑘,𝑡^∗].
Case  1. Let 𝑡=inf{𝑡∈(𝑡_𝑘,𝑡^∗],ℎ(𝑡,𝑥(𝑡))≥𝜌}. As ℎ(𝑡⁺_𝑘,𝑥⁺_𝑘)<𝜌₁<𝜌, we know that 𝑡_𝑘<𝑡≤𝑡^∗. If 𝑡 is left-dense, from the selection of 𝑡, we know that there exists a left-hand neighborhood 𝑈_𝜖=(𝑡−𝜖,𝑡)⊂(𝑡_𝑘,𝑡^∗] for some 𝜖>0, such that 𝜌₁<ℎ(𝑡,𝑥(𝑡))<𝜌 for all 𝑡∈𝑈_𝜖. Then, we can choose 𝑡⁰∈𝑈_𝜖.
If 𝑡 is left-scattered, from the selection of 𝑡 and ℎ(𝑡⁺_𝑘,𝑥⁺_𝑘)<𝜌₁, we know that 𝑡_𝑘<𝜃(𝑡)<𝑡^∗ and ℎ(𝜃(𝑡),𝑥(𝜃(𝑡)))<𝜌. Here, we claim that ℎ(𝜃(𝑡),𝑥(𝜃(𝑡)))≥𝜌₁. If this is not true, that is, ℎ(𝜃(𝑡),𝑥(𝜃(𝑡)))<𝜌₁, form condition (v), we know that ℎ𝑡,𝑥𝑡=ℎ𝜎𝜃𝑡,𝑥𝜎𝜃𝑡<𝜌, (26) which is a contradiction. Thus, 𝜌₁≤ℎ(𝜃(𝑡),𝑥(𝜃(𝑡)))<𝜌. Then, we can choose 𝑡⁰=𝜃(𝑡).
Case  2. If ℎ(𝑡,𝑥(𝑡))<𝜌 for all 𝑡∈(𝑡_𝑘,𝑡^∗], then we can choose 𝑡⁰=𝑡^∗.
Hence, we can find a 𝑡⁰, 𝑡_𝑘<𝑡⁰≤𝑡^∗, such that (25) holds.
For 𝑡₀≤𝑡≤𝑡⁰ and by conditions (ii) and (iii), we obtain 𝐷⁺𝑉^Δ(𝑡,𝑥)≤𝑐_𝑖𝑉(𝑡,𝑥),𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑡≤𝑡⁰,𝑉𝑡⁺_𝑖,𝑥_𝑖+𝐼_𝑖𝑥_𝑖≤𝜍_𝑖𝑉𝑡_𝑖,𝑥_𝑖,𝑡₀<𝑡_𝑖<𝑡⁰. (27)(28) By (27), we will show that 𝑉(𝑡,𝑥)≤𝑒_𝑐0𝑡,𝑡₀𝑉𝑡₀,𝑥₀,𝑡∈𝑡₀,𝑡₁. (29) To do this, we apply the induction principle ([17], Theorem 1.7) on [𝑡₀,𝑡₁] to the statement 𝐴(𝑡)∶𝑉(𝑡,𝑥)≤𝑒_𝑐0𝑡,𝑡₀𝑉𝑡₀,𝑥₀. (30)
(1) The statement 𝐴(𝑡₀) is true since 𝑒_𝑐0(𝑡₀,𝑡₀)𝑉(𝑡₀,𝑥₀)=𝑉(𝑡₀,𝑥₀).
(2) Let 𝑡 be rs and 𝐴(𝑡) be true. We have to prove that 𝐴(𝜎(𝑡)) is true.
By the definition of upper right-hand derivative, we see that 𝐷⁺𝑉^Δ()=𝑉(𝜎(𝑡)(𝜎(𝑡((𝑡𝑡,𝑥,𝑥)))−𝑉𝑡,𝑥))𝜇(𝑡)≤𝑐₀𝑉(𝑡,𝑥), (31) then 𝑉(𝜎(𝑡),𝑥(𝜎(𝑡)))≤1+𝑐₀𝜇(𝑡)𝑉(𝑡,𝑥)=𝑒_𝑐0(𝜎(𝑡),𝑡)𝑉(𝑡,𝑥)≤𝑒_𝑐0(𝜎(𝑡),𝑡)𝑒_𝑐0𝑡,𝑡₀𝑉𝑡₀,𝑥₀=𝑒_𝑐0𝜎(𝑡),𝑡₀𝑉𝑡₀,𝑥₀ (32) which implies that 𝐴(𝜎(𝑡)) is true.
(3) Let 𝑡 be rd, 𝐴(𝑡) be true and 𝑁 be a neighborhood of 𝑡. We need to show that 𝐴(𝑠) is true for s>𝑡, 𝑠∈𝑁. By (27) and Remark 6, we get 𝑉(𝑠,𝑥(𝑠))≤𝑒^{𝑐0(𝑠−𝑡)}𝑉(𝑡,𝑥)≤𝑒^{𝑐0(𝑠−𝑡)}𝑒_𝑐0𝑡,𝑡₀𝑉𝑡₀,𝑥₀=𝑒_𝑐0(𝑠,𝑡)𝑒_𝑐0𝑡,𝑡₀𝑉𝑡₀,𝑥₀=𝑒_𝑐0𝑠,𝑡₀𝑉𝑡₀,𝑥₀ (33) which implies that 𝐴(𝑠) is true.
(4) Let 𝑡 be ld and 𝐴(𝑠) be true for all 𝑠<𝑡. We need to show that 𝐴(𝑡) is true. By the continuous property of function 𝑉 and the exponential function, it follows that 𝑉(𝑡,𝑥)=lim_{𝑠→𝑡−}𝑉(𝑠,𝑥(𝑠))≤lim_{𝑠→𝑡−}𝑒_𝑐0𝑠,𝑡₀𝑉𝑡₀,𝑥₀=𝑒_𝑐0𝑡,𝑡₀𝑉𝑡₀,𝑥₀ (34) which implies that 𝐴(𝑡) is true.
Hence, we conclude that (29) is true.
Similarly, we can prove that 𝑉(𝑡,𝑥)≤𝑒_𝑐𝑖𝑡,𝑡_𝑖𝑉𝑡⁺_𝑖,𝑥𝑡⁺_𝑖,for𝑡∈𝑡_𝑖,𝑡_𝑖+1,𝑡≤𝑡⁰. (35) Then, by (28), (35), and (20), we obtain 𝑉𝑡⁰,𝑥𝑡⁰≤𝑒_𝑐𝑘𝑡⁰,𝑡_𝑘𝑉𝑡⁺_𝑘,𝑥𝑡⁺_𝑘≤𝑒_𝑐𝑘𝑡⁰,𝑡_𝑘𝜍_𝑘𝑉𝑡_𝑘,𝑥𝑡_𝑘≤𝑒_𝑐𝑘𝑡⁰,𝑡_𝑘𝜍_𝑘𝑒_𝑐𝑘−1𝑡_𝑘,𝑡_𝑘−1𝑉𝑡⁺_𝑘−1,𝑥𝑡⁺_𝑘−1≤⋯≤𝑒_𝑐𝑘𝑡⁰,𝑡_𝑘𝜍_𝑘𝑒_𝑐𝑘−1𝑡_𝑘,𝑡_𝑘−1𝜍_𝑘−1⋯𝑒_𝑐1𝑡₂,𝑡₁×𝜍₁𝑒_𝑐0𝑡₁,𝑡₀𝑉𝑡₀,𝑥₀≤^𝑘_𝑖=0𝜍_𝑖𝑒_𝑐𝑖𝑡_𝑖+1,𝑡_𝑖𝑉𝑡₀,𝑥₀≤𝑀𝑉𝑡₀,𝑥₀≤𝑀𝑎𝑡₀,ℎ₀𝑡₀,𝑥₀≤𝑀𝑎𝑡₀,𝛿<𝑏(𝜀) (36) that is, 𝑉(𝑡⁰,𝑥(𝑡⁰))<𝑏(𝜀). Thus, by (18) and (25), 𝑏(𝜀)≤𝑏ℎ𝑡⁰,𝑥𝑡⁰≤𝑉𝑡⁰,𝑥𝑡⁰<𝑏(𝜀) (37) which is a contradiction. Therefore (22) is true and system (10) is (ℎ₀,ℎ)-stable.

Theorem 13. Assume that all conditions of Theorem 12 hold with the following changes:  (i)^∗ℎ₀ is finer than ℎ;  (ii)^∗𝑉(𝑡,𝑥) is ℎ₀-decrescent.
Then, system (10) is (ℎ₀,ℎ)-uniformly stable.

Proof. From conditions (i)* and (ii)*, the number 𝛿 in the proof of Theorem 12 can be chosen independent of 𝑡₀. Then following the same reasoning of Theorem 12, we can get the (ℎ₀,ℎ)-uniform stability of system (10). The details are omitted.

If 𝑐_𝑘≡0 in condition (ii) of the previous theorems, then the Lyapunov function 𝑉 is monotone along the solutions of system (10) in each impulsive intervals. In this case, we have the following conservative result.

Corollary 14. Assume that (i)ℎ₀,ℎ∈Γ, 𝑉∈𝜈₀, 𝑉(𝑡,𝑥) is ℎ-positive definite on 𝑆(ℎ,𝜌), locally Lipschtiz in 𝑥 for each 𝑡∈𝕋 which is rd, and 𝐷⁺𝑉^Δ(𝑡,𝑥)≤0 for 𝑡≠𝑡_𝑘 and (𝑡,𝑥)∈𝑆(ℎ,𝜌); (ii)𝑉(𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘(𝑥_𝑘))−𝑉(𝑡_𝑘,𝑥_𝑘)≤𝑑_𝑘𝑉(𝑡_𝑘,𝑥_𝑘) for (𝑡_𝑘,𝑥_𝑘)∈𝑆(ℎ,𝜌), where 𝑑_𝑘≥0 and ∑^∞_𝑖=1𝑑_𝑖<∞,
and conditions (v), (vi) of Theorem 12 hold. Then (A)if, in addition, ℎ₀ is weakly finer than ℎ, and 𝑉(𝑡,𝑥) is ℎ₀-weakly decrescent, then system (10) is (ℎ₀,ℎ)-stable;(B)if, in addition, ℎ₀ is finer than ℎ, and 𝑉(𝑡,𝑥) is ℎ₀-decrescent, then system (10) is (ℎ₀,ℎ)-uniformly stable.

Proof. Notice sup_𝑘∈ℤ+^𝑘_𝑖=0𝜍_𝑖𝑒_𝑐𝑖𝑡_𝑖+1,𝑡_𝑖=sup_𝑘∈ℤ+^𝑘_𝑖=01+𝑑_𝑖≤^∞_𝑖=11+𝑑_𝑖<∞, (38) where 𝑑₀=0. Then, by Theorems 12 and 13, the result holds.

Remark 15. The continuous version of Corollary 14 with 𝑑_𝑘≡0, 𝑘∈ℕ, can be found in [7], while the discrete one for impulsive discrete systems is brand new, and the discrete version of Corollary 14 with 𝑉(𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘(𝑥_𝑘))=𝑉(𝑡_𝑘,𝑥(𝑡_𝑘)), 𝑘∈ℕ, reduces to Theorems 3.1 and 3.2 in [18] for discrete systems with no impulses.

4.2. (ℎ₀,ℎ)-(Uniform) Asymptotic Stability

Theorem 16. Assume that conditions (v), (vi) of Theorem 12 and condition (i) of Corollary 14 hold and the following conditions are satisfied: (i)ℎ₀ is weakly finer than ℎ, and 𝑉(𝑡,𝑥) is ℎ₀-weakly decrescent; (ii)𝑉(𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘(𝑥_𝑘))−𝑉(𝑡_𝑘,𝑥_𝑘)≤−𝜆_𝑘𝜓(𝑉(𝑡_𝑘,𝑥_𝑘)) for (𝑡_𝑘,𝑥_𝑘)∈𝑆(ℎ,𝜌), where 𝜆_𝑘≥0, ∑^∞_𝑘=1𝜆_𝑘=∞, 𝜓∶ℝ⁺→ℝ⁺, 𝜓(0)=0 and 𝜓(𝑠)>0 if 𝑠>0.
Then system (10) is (ℎ₀,ℎ)-asymptotically stable.

Proof. By Theorem 12, system (10) is (ℎ₀,ℎ)-stable. Thus, for 𝜌>0, there exists 𝛿=𝛿(𝑡₀,𝜌)>0 such that ℎ₀(𝑡₀,𝑥₀)<𝛿 implies ℎ(𝑡,𝑥)<𝜌, 𝑡≥𝑡₀. To prove the theorem, it remains to show that lim_𝑡→∞ℎ(𝑡,𝑥(𝑡))=0.
Let 𝑚(𝑡)=𝑉(𝑡,𝑥(𝑡)). Then it follows from assumptions that 𝑚(𝑡) is right-nonincreasing and bounded from below, and consequently lim_𝑡→∞𝑚(𝑡)=𝜔≥0 exists. If 𝜔>0 for some solution 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) of (10), we let 𝛾=min_{𝜔≤𝑠≤𝑚(𝑡0)}𝜓(𝑠). Then, by condition (ii), we have 𝑚𝑡⁺_𝑘−𝑚𝑡_𝑘≤−𝜆_𝑘𝜓𝑚𝑡_𝑘≤−𝜆_𝑘𝛾. (39) Thus we obtain from (39) that 𝑚𝑡⁺_𝑘≤𝑚𝑡_𝑘−𝜆_𝑘𝛾≤𝑚𝑡⁺_𝑘−1−𝜆_𝑘𝛾≤𝑚𝑡_𝑘−1−𝜆_𝑘−1𝛾−𝜆_𝑘𝛾𝑡≤⋯≤𝑚₁−𝛾^𝑘_𝑗=1𝜆_𝑗, (40) which implies, in view of the assumption ∑^𝑘_𝑗=1𝜆_𝑗=∞, that lim_𝑘→∞𝑚(𝑡⁺_𝑘)=−∞. This is a contradiction. Thus we must have 𝜔=0 and consequently lim_𝑡→∞ℎ(𝑡,𝑥(𝑡))=0. Hence system (10) is (ℎ₀,ℎ)-attractive and the proof is complete.

In the following theorems, two auxiliary functions of class 𝜈₀ are used to investigate the (ℎ₀,ℎ)-asymptotic stability property of system (10).

Theorem 17. Let conditions (v), (vi) of Theorem 12 hold and assume that (i)ℎ₀,ℎ∈Γ and ℎ₀ is weakly finer than ℎ; (ii) there exists a function 𝑉∈𝜈₀ such that 𝑉(𝑡,𝑥) is locally Lipschtiz in 𝑥 for each 𝑡∈𝕋 which is rd, ℎ-positive definite on 𝑆(ℎ,𝜌), ℎ₀-weakly decrescent and 𝐷⁺𝑉^Δ(𝑡,𝑥)≤−𝑐(𝑊(𝑡,𝑥)),𝑡≠𝑡_𝑘,(𝑡,𝑥)∈𝑆ℎ,𝜌, (41) where 𝑐∈𝒦, 𝑊∈𝜈₀; (iii)𝑉(𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘(𝑥_𝑘))−𝑉(𝑡_𝑘,𝑥_𝑘)≤𝑑_𝑘𝑉(𝑡_𝑘,𝑥_𝑘), (𝑡_𝑘,𝑥_𝑘)∈𝑆(ℎ,𝜌), where 𝑑_𝑘≥0 and ∑^∞_𝑖=1𝑑_𝑖<∞; (iv)𝑊(𝑡,𝑥) is ℎ-positive definite on 𝑆(ℎ,𝜌), locally Lipschtiz in 𝑥 for every 𝑡∈𝕋 which is rd and 𝐷⁺𝑊^Δ(𝑡,𝑥)≤0,𝑡≠𝑡_𝑘,(),𝑊𝑡𝑡,𝑥∈𝑆ℎ,𝜌_+𝑘,𝑥_𝑘+𝐼_𝑘𝑥_𝑘−𝑊𝑡_𝑘,𝑥_𝑘≤𝑑_𝑘𝑊𝑡_𝑘,𝑥_𝑘,𝑡_𝑘,𝑥_𝑘∈𝑆ℎ,𝜌, (42) where 𝑑_𝑘≥0 and ∑^∞_𝑖=1𝑑_𝑖<∞.
Then system (10) is (ℎ₀,ℎ)-asymptotically stable.

Proof. From Corollary 14, it follows that system (10) is (ℎ₀,ℎ)-stable. Thus, for 𝜌>0, there exists 𝛿=𝛿(𝑡₀,𝜌)>0 such that ℎ₀(𝑡₀,𝑥₀)<𝛿 implies ℎ(𝑡,𝑥)<𝜌, 𝑡≥𝑡₀. To prove the theorem, it remains to show that for every solution 𝑥(𝑡) of (10) with ℎ₀(𝑡₀,𝑥₀)<𝛿, lim_𝑡→∞ℎ(𝑡,𝑥)=0.
Suppose that this is not true. Then there exists a sequence {𝜉_𝑖}^∞_𝑖=1 diverging to ∞ as 𝑖→∞ and such that ℎ(𝜉_𝑖,𝑥(𝜉_𝑖))≥𝑟(𝑖∈ℕ) for some positive number 𝑟. From condition (iv), we know that there exists a function 𝑏∈𝒦 such that 𝑏(ℎ(𝑡,𝑥))≤𝑊(𝑡,𝑥), if ℎ(𝑡,𝑥)<𝜌. Then 𝑊𝜉_𝑖,𝑥𝜉_𝑖>𝑏(𝑟),𝑖∈ℕ. (43)
For any given 𝑡∈𝕋, there exists a 𝑖>0 such that 𝑡∈(𝜉_𝑖,𝜉_𝑖+1]. Then for 𝜉_𝑖+1, there exists a 𝑘>0 such that 𝜉_𝑖+1∈(𝑡_𝑘,𝑡_𝑘+1]. If 𝑡_𝑘<𝑡≤𝑡_𝑘+1, from (43) and condition (iv), we have 𝑊(𝑡,𝑥)≥𝑊𝜉_𝑖+1,𝑥𝜉_𝑖+1(𝑟)≥𝑏(𝑟)>𝑏𝑀₀, (44) where 𝑀₀=∏^∞_𝑗=1(1+𝑑_𝑗)<∞. If 𝑡_𝑘−1<𝑡≤𝑡_𝑘, we have 𝑊(𝑡,𝑥)≥𝑊𝑡_𝑘,𝑥_𝑘≥11+𝑑_𝑘𝑊𝑡⁺_𝑘,𝑥𝑡⁺_𝑘>11+𝑑_𝑘𝑏(𝑟)≥𝑏(𝑟)𝑀₀. (45)
Following this procedure, we conclude that 𝑊()>𝑏(𝑟)𝑡,𝑥𝑀₀,𝑡∈𝕋. (46)
Let 𝐿(𝑡,𝑥)=𝑉(𝑡,𝑥)+^𝑡_𝑡𝑘𝑐(𝑊((𝑠𝑡𝑠,𝑥)))Δ𝑠,𝑡∈_𝑘,𝑡_𝑘+1,𝑘∈ℤ⁺. (47) Then, by condition (ii), 𝐷⁺𝐿^Δ(𝑡,𝑥)=𝐷⁺𝑉^Δ(𝑡,𝑥)+𝑐(𝑊(𝑡,𝑥))≤0,𝑡≠𝑡_𝑘, (48) which implies 𝑉(𝑡,𝑥)≤𝑉𝑡⁺_𝑘,𝑥𝑡⁺_𝑘−^𝑡_𝑡𝑘𝑐(𝑊((𝑠𝑡𝑠,𝑥)))Δ𝑠,𝑡∈_𝑘,𝑡_𝑘+1, (49) for 𝑘∈ℤ⁺.
Hence, for 𝜉_𝑖+1∈(𝑡_𝑘,𝑡_𝑘+1], we obtain, form (46), (49), and condition (iii), 𝑉𝜉_𝑖+1,𝑥𝜉_𝑖+1≤𝑉𝑡⁺_𝑘,𝑥𝑡⁺_𝑘−^𝜉𝑖+1_𝑡𝑘𝑐(𝑊((𝑠≤𝑠,𝑥)))Δ𝑠1+𝑑_𝑘𝑉𝑡_𝑘,𝑥𝑡_𝑘−^𝜉𝑖+1_𝑡𝑘𝑐(𝑊((𝑠≤𝑠,𝑥)))Δ𝑠1+𝑑_𝑘𝑉𝑡⁺_𝑘−1,𝑥𝑡⁺_𝑘−1−^𝑡𝑘_𝑡𝑘−1𝑐(𝑊((𝑠−𝑠,𝑥)))Δ𝑠_{𝜉𝑖+1𝑡𝑘}𝑐(𝑊((𝑠≤𝑠,𝑥)))Δ𝑠1+𝑑_𝑘1+𝑑_𝑘−1𝑉𝑡_𝑘−1,𝑥𝑡_𝑘−1−^𝜉𝑖+1_𝑡𝑘−1𝑐(𝑊((𝑠≤𝑠,𝑥)))Δ𝑠≤⋯_𝑘𝑗=11+𝑑_𝑗𝑉𝑡₀,𝑥₀−^𝜉𝑖+1_𝑡0𝑐(𝑊((𝑠𝑡𝑠,𝑥)))Δ𝑠≤𝑀𝑉₀,𝑥₀𝑏(𝑟)−𝑐𝑀₀𝜉_𝑖+1−𝑡₀→−∞,for𝑖→∞, (50) where 𝑀=∏^∞_𝑗=1(1+𝑑_𝑗)<∞. This is a contradiction, hence lim_𝑡→∞ℎ(𝑡,𝑥)=0. Theorem 17 is proved.

In Theorem 17, the function may have a special form. In the case when 𝑊(𝑡,𝑥)=𝑉(𝑡,𝑥) and 𝑑_𝑘=𝑑_𝑘, 𝑘∈ℕ, we deduce the following corollary.

Corollary 18. Let conditions (v), (vi) of Theorem 12 hold and assume that (i)ℎ₀,ℎ∈Γ and ℎ₀ is weakly finer than ℎ; (ii)there exist 𝑐∈𝒦, and function 𝑉∈𝜈₀ such that 𝑉(𝑡,𝑥) is locally Lipschtiz in 𝑥 for each 𝑡∈𝕋 which is rd, ℎ-positive definite on 𝑆(ℎ,𝜌), ℎ₀-weakly decrescent and 𝐷⁺𝑉^Δ(𝑡,𝑥)≤−𝑐(𝑉(𝑡,𝑥)),𝑡≠𝑡_𝑘,(𝑡,𝑥)∈𝑆ℎ,𝜌; (51)(iii)𝑉(𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘(𝑥_𝑘))−𝑉(𝑡_𝑘,𝑥_𝑘)≤𝑑_𝑘𝑉(𝑡_𝑘,𝑥_𝑘), (𝑡_𝑘,𝑥_𝑘)∈𝑆(ℎ,𝜌), where 𝑑_𝑘≥0 and ∑^∞_𝑖=1𝑑_𝑖<∞.
Then system (10) is (ℎ₀,ℎ)-asymptotically stable.

Theorem 19. Assume that all conditions of Theorem 13 hold. Suppose further, that there exists a function 𝑊∈𝜈₀ such that 𝑊(𝑡,𝑥) is locally Lipschtiz in 𝑥 for each 𝑡∈𝕋 which is rd, and the following conditions hold: (i)𝐷⁺𝑊^Δ(𝑡,𝑥)≤−𝑝(𝑡)𝑐(ℎ₀(𝑡,𝑥))+𝑞(𝑡), 𝑡≠𝑡_𝑘, (𝑡,𝑥)∈𝑆(ℎ,𝜌), where 𝑐∈𝒦, 𝑝,𝑞∈𝐶_𝑟𝑑(𝕋,ℝ⁺), ∫^∞_𝑡0𝑝(𝜏)Δ𝜏=∞, and ∫^∞_𝑡0𝑞(𝜏)Δ𝜏<∞; (ii)𝑊(𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘(𝑥_𝑘))−𝑊(𝑡_𝑘,𝑥_𝑘)≤𝑑_𝑘𝑊(𝑡_𝑘,𝑥_𝑘), (𝑡_𝑘,𝑥_𝑘)∈𝑆(ℎ,𝜌), where 𝑑_𝑘≥0 and ∑^∞_𝑖=1𝑑_𝑖<∞.
Then system (10) is (ℎ₀,ℎ)-attractive.

Proof. By Theorem 13, system (10) is (ℎ₀,ℎ)-uniformly stable. Thus, for 𝜌>0, there exists a 𝛿₀=𝛿₀(𝜌)>0 such that ℎ₀(𝑡₀,𝑥₀)<𝛿₀ implies that ℎ(𝑡,𝑥)<𝜌, 𝑡≥𝑡₀, where 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) is any solution of (10).
Let 𝜀∈(0,𝜌) be given, 𝛿=𝛿(𝜀)>0 be the same as defined in the definition of (ℎ₀,ℎ)-uniform stability, and ℎ₀(𝑡₀,𝑥₀)<𝛿₀. We claim that there exists a 𝑡^∗≥𝑡₀ such that ℎ₀𝑡^∗,𝑥𝑡^∗<𝛿. (52) If this is not true, then ℎ₀(𝑡,𝑥)≥𝛿 for all 𝑡≥𝑡₀.
Let 𝐿(𝑡,𝑥)=𝑊(𝑡,𝑥)+^𝑡_𝑡𝑘𝑝(𝜏)𝑐ℎ₀((𝜏−𝜏,𝑥))Δ𝜏_𝑡𝑡𝑘𝑞(𝜏)Δ𝜏,𝑡∈𝑡_𝑘,𝑡_𝑘+1, (53) for 𝑘∈ℤ⁺. By condition (i), we obtain 𝐷⁺𝐿^Δ(𝑡,𝑥)=𝐷⁺𝑊^Δ(𝑡,𝑥)+𝑝(𝑡)ℎ₀(𝑡,𝑥)−𝑞(𝑡)≤0,𝑡≠𝑡_𝑘, (54) which implies that, for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘∈ℤ⁺, 𝐿(𝑡,𝑥)≤𝐿𝑡⁺_𝑘,𝑥𝑡⁺_𝑘. (55) Then, it follows from (55) and condition (ii) that, for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑊(𝑡,𝑥)≤𝑊𝑡⁺_𝑘,𝑥𝑡⁺_𝑘−^𝑡_𝑡𝑘𝑝(𝜏)𝑐ℎ₀((𝜏+𝜏,𝑥))Δ𝜏_𝑡𝑡𝑘𝑞(𝜏)≤Δ𝜏1+𝑑_𝑘𝑊𝑡_𝑘,𝑥𝑡_𝑘−^𝑡_𝑡𝑘𝑝(𝜏)𝑐ℎ₀(𝜏,𝑥(𝜏))Δ𝜏+^𝑡_𝑡𝑘𝑞(𝜏)≤Δ𝜏1+𝑑_𝑘×𝑊𝑡⁺_𝑘−1,𝑥𝑡⁺_𝑘−1−^𝑡𝑘_𝑡𝑘−1𝑝(𝜏)𝑐ℎ₀((𝜏+𝜏,𝑥))Δ𝜏_{𝑡𝑘𝑡𝑘−1}𝑞(𝜏)Δ𝜏−^𝑡_𝑡𝑘𝑝(𝜏)𝑐ℎ₀(𝜏,𝑥(𝜏))Δ𝜏+^𝑡_𝑡𝑘𝑞(𝜏)≤Δ𝜏1+𝑑_𝑘1+𝑑_𝑘−1𝑊𝑡_𝑘−1,𝑥𝑡_𝑘−1−^𝑡_𝑡𝑘−1𝑝(𝜏)𝑐ℎ₀((𝜏+𝜏,𝑥))Δ𝜏1+𝑑_𝑘^𝑡_𝑡𝑘−1𝑞(𝜏)≤Δ𝜏≤⋯_𝑘𝑖=11+𝑑_𝑖𝑊𝑡₀,𝑥₀−^𝑡_𝑡0𝑝(𝜏)𝑐ℎ₀((𝜏+𝜏,𝑥))Δ𝜏_𝑘𝑖=21+𝑑_𝑖^𝑡_𝑡0𝑞(𝜏)Δ𝜏≤𝑀₀𝑊𝑡₀,𝑥₀+^𝑡_𝑡0𝑞(𝜏)Δ𝜏−𝑐(𝛿)^𝑡_𝑡0𝑝(𝜏)Δ𝜏, (56) where 𝑀₀=∏^∞_𝑖=1(1+𝑑_𝑖)<∞. Then, (56) implies that 𝑊(𝑡,𝑥)→−∞, for 𝑡→∞. This contradiction shows that (52) is true, and hence, ℎ(𝑡,𝑥)<𝜀,𝑡≥𝑡^∗. (57) Thus we conclude that system (10) is (ℎ₀,ℎ)-attractive.

Remark 20. When 𝕋=ℤ, and 𝑑_𝑘=𝑑_𝑘=0, 𝑘∈ℕ, Theorem 19 contains Theorem  3.4 in [18] for discrete systems without impulse effects.
Next, we will give two results on uniform asymptotic stability in terms of two measures.

Theorem 21. Let all the conditions of Theorem 19 and the following additional conditions hold: (i)𝑊(𝑡,𝑥) is ℎ₀-decrescent, and 𝑝(𝑡)≡𝑝  (𝑝 is a positive constant), for 𝑡∈𝕋; (ii)there exists a constant 𝜏>0 such that {𝑡+𝑘𝜏∶𝑡∈𝕋,𝑘∈ℕ}⊂𝕋. (58)
Then system (10) is uniformly asymptotically stable.

Proof. Since 𝑊(𝑡,𝑥) is ℎ₀-decrescent, there exist 𝛿₁>0 and a function 𝑏∈𝒦 such that 𝑊(𝑡,𝑥)≤𝑏ℎ₀(𝑡,𝑥),ifℎ₀(𝑡,𝑥)<𝛿₁. (59) By Theorem 19, system (10) is (ℎ₀,ℎ)-uniformly stable. Thus, there exists a 𝛿₀=𝛿₀(𝜌)∈(0,𝛿₁) such that ℎ₀(𝑡₀,𝑥₀)<𝛿₀ implies ℎ(𝑡,𝑥)<𝜌, 𝑡≥𝑡₀, for any solution 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) of (10).
Let 𝜀∈(0,𝜌) be given and 𝛿=𝛿(𝜀)>0 be the same as defined in the definition of (ℎ₀,ℎ)-uniformly stability. Let 𝑚>0 be the smallest integer such that 𝑀𝑚>⁰𝑏𝛿₀+𝑁(𝛿),𝑝𝜏𝑐 (60) where 𝑀₀=∏^∞_𝑘=1(1+𝑑_𝑘)<∞ and 𝑁=∫^∞_𝑡0𝑞(𝑠)Δ𝑠<∞.
Choose 𝑇=𝑚𝜏 and let 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) be any solution of (10) with ℎ₀(𝑡₀,𝑥₀)<𝛿₀. We claim that there exists a 𝑡^∗∈[𝑡₀,𝑡₀+𝑇] such that ℎ₀(𝑡^∗,𝑥(𝑡^∗))<𝛿. If this is not true, then ℎ₀(𝑡,𝑥)≥𝛿 for all 𝑡^∗∈[𝑡₀,𝑡₀+𝑇]. By (56), (60), and condition (ii), we have 𝑊𝑡₀+𝑇,𝑥𝑡₀+𝑇≤𝑀₀𝑊𝑡₀,𝑥₀−𝑝𝑐(𝛿)𝑇+𝑀₀^𝑡0_𝑡^+𝑇₀𝑞(𝑠)Δ𝑠≤𝑀₀𝑏𝛿₀−𝑚𝑐(𝛿)𝑝𝜏+𝑀₀𝑁<0 (61) which is a contradiction. Thus, our claim is true and by the uniform stability we have ℎ(𝑡,𝑥)<𝜀,𝑡≥𝑡₀+𝑇≥𝑡^∗. (62) Hence, system (10) is (ℎ₀,ℎ)-uniformly attractive. This completes the proof.

Theorem 22. Let conditions (v), (vi) of Theorem 12 hold and assume that (i)ℎ₀,ℎ∈Γ and ℎ₀ is finer than ℎ; (ii)there exist a 𝑐∈𝒦, and function 𝑉∈𝜈₀ such that 𝑉(𝑡,𝑥) is locally Lipschtiz in 𝑥 for each 𝑡∈𝕋 which is rd, ℎ-positive definite on 𝑆(ℎ,𝜌), ℎ₀-decrescent and 𝐷⁺𝑉^Δ(𝑡,𝑥)≤−𝑐ℎ₀(𝑡,𝑥),𝑡≠𝑡_𝑘,(𝑡,𝑥)∈𝑆ℎ,𝜌; (63)(iii)𝑉(𝑡⁺_𝑘,𝑥_𝑘+𝐼_𝑘(𝑥_𝑘))−𝑉(𝑡_𝑘,𝑥_𝑘)≤𝑑_𝑘𝑉(𝑡_𝑘,𝑥_𝑘), (𝑡_𝑘,𝑥_𝑘)∈𝑆(ℎ,𝜌), where 𝑑_𝑘≥0 and ∑^∞_𝑖=1𝑑_𝑖<∞.
Then system (10) is (ℎ₀,ℎ)-uniformly asymptotically stable.

Proof. Since 𝑉(𝑡,𝑥) is ℎ₀-decrescent, there exist a constant 𝛿₀ and a function 𝑎∈𝒦 such that 𝑉(𝑡,𝑥)≤𝑎ℎ₀(𝑡,𝑥),ifℎ₀(𝑡,𝑥)<𝛿₀. (64) The fact that 𝑉(𝑡,𝑥) is ℎ-positive definite on 𝑆(ℎ,𝜌) implies that there exists a function 𝑏∈𝒦 such that 𝑏(ℎ(𝑡,𝑥))≤𝑉(𝑡,𝑥),ifℎ(𝑡,𝑥)<𝜌. (65)
It follows from Corollary 14 that system (10) is (ℎ₀,ℎ)-uniformly stable. Thus for 𝜌>0, there exist a 𝛿₁=𝛿₁(𝜌)∈(0,𝛿₀) such that ℎ₀(𝑡₀,𝑥₀)<𝛿₁ implies ℎ(𝑡,𝑥)<𝜌,𝑡≥𝑡₀. (66) By the choice of 𝛿₁, we get 𝑉𝑡₀,𝑥₀≤𝑎ℎ₀𝑡₀,𝑥₀≤𝑎𝛿₁. (67)
To prove the theorem, it is enough to show that system (10) is (ℎ₀,ℎ)-uniformly attractive.
Given 0<𝜀<𝜌, let 𝛿=𝛿(𝜀)>0 be the same as defined in the definition of (ℎ₀,ℎ)-uniformly stability. Then for any solution 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) of system (10) with ℎ₀(𝑡₀,𝑥₀)<𝛿₁, we claim that there exists a 𝑇=𝑇(𝜀)>𝑀𝑎(𝛿₁)/𝑐(𝛿) such that, for some 𝑡^∗∈[𝑡₀,𝑡₀+𝑇], ℎ₀𝑡^∗,𝑥𝑡^∗<𝛿, (68) where 𝑀=∏^∞_𝑘=1(1+𝑑_𝑘)<∞. Suppose that this is false. Then for any 𝑇>𝑀𝑎(𝛿₁)/𝑐(𝛿) there exists a solution 𝑥(𝑡)=𝑥(𝑡;𝑡₀,𝑥₀) of (10) satisfying ℎ₀(𝑡₀,𝑥₀)<𝛿₁, such that ℎ₀(𝑡,𝑥(𝑡))≥𝛿,𝑡∈𝑡₀,𝑡₀+𝑇. (69)
By setting 𝐿(𝑡,𝑥)=𝑉(𝑡,𝑥)+^𝑡_𝑡𝑘𝑐ℎ₀((𝑠𝑡𝑠,𝑥))Δ𝑠,𝑡∈_𝑘,𝑡_𝑘+1,𝑘∈ℤ⁺, (70) and condition (ii), we have 𝐷⁺𝐿^Δ(𝑡,𝑥)=𝐷⁺𝑉^Δ(𝑡,𝑥)+𝑐ℎ₀(𝑡,𝑥)≤0,𝑡≠𝑡_𝑘, (71) which implies 𝐿(𝑡,𝑥)≤𝐿(𝑡⁺_𝑘,𝑥(𝑡⁺_𝑘)), for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1]. Then, for 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], we get 𝑉(𝑡,𝑥)≤𝑉𝑡⁺_𝑘,𝑥𝑡⁺_𝑘−^𝑡_𝑡𝑘𝑐₀(𝑠)≤Δ𝑠1+𝑑_𝑘𝑉𝑡_𝑘,𝑥𝑡_𝑘−^𝑡_𝑡𝑘𝑐₀(𝑠)≤Δ𝑠1+𝑑_𝑘𝑉𝑡⁺_𝑘−1,𝑥𝑡⁺_𝑘−1−^𝑡𝑘_𝑡𝑘−1𝑐₀(𝑠)−Δ𝑠_𝑡𝑡𝑘𝑐₀(𝑠)≤Δ𝑠1+𝑑_𝑘1+𝑑_𝑘−1𝑉𝑡⁺_𝑘−1,𝑥𝑡⁺_𝑘−1−^𝑡_𝑡𝑘−1𝑐₀(𝑠)𝑡Δ𝑠